Optimal. Leaf size=82 \[ \frac {x}{6 \sqrt {x^2} \left (1-x^2\right )}-\frac {\sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}-\frac {\sec ^{-1}(x)}{\sqrt {-1+x^2}}-\frac {2 x \log (x)}{3 \sqrt {x^2}}+\frac {x \log \left (-1+x^2\right )}{3 \sqrt {x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.02, number of steps
used = 5, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 45, 5346,
12, 457, 78} \begin {gather*} \frac {x}{6 \sqrt {x^2} \left (1-x^2\right )}-\frac {2 x \log (x)}{3 \sqrt {x^2}}+\frac {x \log \left (1-x^2\right )}{3 \sqrt {x^2}}-\frac {\sec ^{-1}(x)}{\sqrt {x^2-1}}-\frac {\sec ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 78
Rule 272
Rule 457
Rule 5346
Rubi steps
\begin {align*} \int \frac {x^3 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx &=-\frac {\sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}-\frac {\sec ^{-1}(x)}{\sqrt {-1+x^2}}-\frac {x \int \frac {2-3 x^2}{3 x \left (1-x^2\right )^2} \, dx}{\sqrt {x^2}}\\ &=-\frac {\sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}-\frac {\sec ^{-1}(x)}{\sqrt {-1+x^2}}-\frac {x \int \frac {2-3 x^2}{x \left (1-x^2\right )^2} \, dx}{3 \sqrt {x^2}}\\ &=-\frac {\sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}-\frac {\sec ^{-1}(x)}{\sqrt {-1+x^2}}-\frac {x \text {Subst}\left (\int \frac {2-3 x}{(1-x)^2 x} \, dx,x,x^2\right )}{6 \sqrt {x^2}}\\ &=-\frac {\sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}-\frac {\sec ^{-1}(x)}{\sqrt {-1+x^2}}-\frac {x \text {Subst}\left (\int \left (-\frac {1}{(-1+x)^2}-\frac {2}{-1+x}+\frac {2}{x}\right ) \, dx,x,x^2\right )}{6 \sqrt {x^2}}\\ &=\frac {x}{6 \sqrt {x^2} \left (1-x^2\right )}-\frac {\sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}-\frac {\sec ^{-1}(x)}{\sqrt {-1+x^2}}-\frac {2 x \log (x)}{3 \sqrt {x^2}}+\frac {x \log \left (1-x^2\right )}{3 \sqrt {x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 72, normalized size = 0.88 \begin {gather*} \frac {-2 \left (-2+3 x^2\right ) \sec ^{-1}(x)-\frac {\left (-1+x^2\right ) \left (1+4 \left (-1+x^2\right ) \log (x)-2 \left (-1+x^2\right ) \log \left (1-x^2\right )\right )}{\sqrt {1-\frac {1}{x^2}} x}}{6 \left (-1+x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.65, size = 197, normalized size = 2.40
method | result | size |
default | \(-\frac {4 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \,\mathrm {arcsec}\left (x \right )}{3 \sqrt {x^{2}-1}}+\frac {\sqrt {x^{2}-1}\, \left (2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -3 x^{2}+2\right ) \left (2 i x^{4}+8 x^{4} \mathrm {arcsec}\left (x \right )+3 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-4 i x^{2}-6 \,\mathrm {arcsec}\left (x \right ) x^{2}-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +2 i\right )}{6 x^{2} \left (4 x^{6}-11 x^{4}+10 x^{2}-3\right )}+\frac {2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \ln \left (\left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )^{2}-1\right )}{3 \sqrt {x^{2}-1}}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 69, normalized size = 0.84 \begin {gather*} -\frac {2 \, {\left (3 \, x^{2} - 2\right )} \sqrt {x^{2} - 1} \operatorname {arcsec}\left (x\right ) + x^{2} - 2 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x^{2} - 1\right ) + 4 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x\right ) - 1}{6 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.75, size = 64, normalized size = 0.78 \begin {gather*} -\frac {{\left (3 \, x^{2} - 2\right )} \arccos \left (\frac {1}{x}\right )}{3 \, {\left (x^{2} - 1\right )}^{\frac {3}{2}}} - \frac {\log \left (x^{2}\right )}{3 \, \mathrm {sgn}\left (x\right )} + \frac {\log \left ({\left | x^{2} - 1 \right |}\right )}{3 \, \mathrm {sgn}\left (x\right )} - \frac {2 \, x^{2} - 1}{6 \, {\left (x^{2} - 1\right )} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\mathrm {acos}\left (\frac {1}{x}\right )}{{\left (x^2-1\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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